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3 years ago

Help please ............ must show work

Mathematics

1 answer:

ehidna [41]3 years ago

7 0

1. 4^2+4^2=n^2

16+16=n^2

32=n^2

n=5.66

2. n^2+8^2=12^2

n^2+64=144

n^2=80

n=8.94

3. 5^2+7^2=n^2

25+49=n^2

74=n^2

n=8.60

4. 5^2+n^2=13^2

63+121=n^2

185=n^2

n=13.60

5.8^2+11^2=n^2

64+121=n^2

185=n^2

n=13.60

6. 7^2+11^2=n^2

49+121=n^2

170=n^2

n=13.04

7.6^2+10^2=n^2

36+100=n^2

136=n^2

n=11.66

8. 9^2+n^2=16^2

81+n^2=256

n^2=175

n=13.23

9.8^2+n^2=12^2

64+n^2=144

n^2=80

n=8.94

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A plausible guess might be that the sequence is formed by a degree-4* polynomial,

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$\begin{cases}a+b+c+d+e = -2 \\ 16 a + 8 b + 4 c + 2 d + e = 1 \\ 81 a + 27 b + 9 c + 3 d + e = 7 \\ 256 a + 64 b + 16 c + 4 d + e = 25 \\ 625 a + 125 b + 25 c + 5 d + e = 79\end{cases}$

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so that the n-th term in the sequence might be

$\displaystyle x_n = \boxed{\frac{5 n^4}{8}-\frac{19 n^3}{4}+\frac{115 n^2}{8}-\frac{65 n}{4}+4}$

Then the next few terms in the sequence could very well be

$\{-2, 1, 7, 25, 79, 208, 466, 922, 1660, 2779, \ldots\}$

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* Why degree-4? This rests on the assumption that the higher-order forward differences of $\{x_n\}$ eventually form a constant sequence. But we only have enough information to find one term in the sequence of 4th-order differences. Denote the k-th-order forward differences of $\{x_n\}$ by $\Delta^{k}\{x_n\}$ . Then

• 1st-order differences:

$\Delta\{x_n\} = \{1-(-2), 7-1, 25-7, 79-25,\ldots\} = \{3,6,18,54,\ldots\}$

• 2nd-order differences:

$\Delta^2\{x_n\} = \{6-3,18-6,54-18,\ldots\} = \{3,12,36,\ldots\}$

• 3rd-order differences:

$\Delta^3\{x_n\} = \{12-3, 36-12,\ldots\} = \{9,24,\ldots\}$

• 4th-order differences:

$\Delta^4\{x_n\} = \{24-9,\ldots\} = \{15,\ldots\}$

From here I made the assumption that $\Delta^4\{x_n\}$ is the constant sequence {15, 15, 15, …}. This implies $\Delta^3\{x_n\}$ forms an arithmetic/linear sequence, which implies $\Delta^2\{x_n\}$ forms a quadratic sequence, and so on up $\{x_n\}$ forming a quartic sequence. Then we can use the method of undetermined coefficients to find it.

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2 years ago

Help please ............ must show work​

Help please ............ must show work